In particular, \(\wedge \)  is equivalent to the exterior derivative \(d\)  on differential forms \(\Omega (M)\) , \(\)  is equivalent to the codifferential \(\delta \) , and \(^2 = \Delta \)  is the Laplace–Beltrami operator. The kernel of \(d+\delta \)  on \(\Omega ^k(M)\)  is coupled to the (co)homology of \(M\)  and so \(\)  retains this property as well. This relationship of the analysis of \(d+\delta \)  (and equivalently \(\) ) to the absolute and relative (co)homology of \(M\)  is formalized in Hodge theory and is useful for proving existence and uniqueness for boundary value problems [1]}. However, in Clifford analysis, it is commonplace to consider mixed grade multivectors also in the kernel of \(\) . This space is far bigger since we find that grades can “mix” together. For example, take \(M\)  to be the unit disk \(\mathbb {D} \subset ^2\)  and \(f_+ = f_0 + f_2 \)  with \(f_0,f_2 \in {\mathbb {D}}{}\)  where \(\)  is the unit 2-blade field. Then we have
\(f_+ = \wedge f_0 + ( f_2 ).\) 
